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Uniform Integrability
In mathematics, uniform integrability is an important concept in real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ..., functional analysis and measure theory, and plays a vital role in the theory of Martingale (probability theory), martingales. Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in L_1 which is central in Dominated convergence theorem, dominated convergence. Several textbooks on real analysis and measure theory use the following definition: Definition A: Let (X,\mathfrak, \mu) be a positive measure space. A set \Phi\subset L^1(\mu) is called uniformly integrable if \sup_\, f\, _0 there corresponds a \delta>0 such that : \int_E , f, \, d\mu 0 such that : \sup_\int_A, f, \, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and , and an order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique ''complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nelson Dunford
Nelson James Dunford (December 12, 1906 – September 7, 1986) was an American mathematician, known for his work in functional analysis, namely integration of vector valued functions, ergodic theory, and linear operators. The Dunford decomposition, Dunford–Pettis property, and Dunford-Schwartz theorem bear his name. He studied mathematics at the University of Chicago and obtained his Ph.D. in 1936 at Brown University under Jacob Tamarkin. He moved in 1939 to Yale University, where he remained until his retirement in 1960. In 1981, he was awarded jointly with Jacob T. Schwartz, his Ph.D. student, the well-known Leroy P. Steele Prize of the American Mathematical Society for the three-volume work ''Linear operators''. Nelson Dunford was coeditor of Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was establi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominated Convergence Theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration. In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables. Statement Lebesgue's dominated convergence theorem. Let (f_n) be a sequence of complex-valued measurable functions on a measure space . Suppose that the sequence converges pointwise to a function f and is dominated by some integrable function g in the sense that : , f_n(x), \le g(x) for all numbers ''n'' in the index set of the sequence and all points x\in S. Then ''f'' is integrable (in the Lebesgue sense) and : \lim_ \int_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence In Probability
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied. The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution. Background "Stochastic convergence" formalizes the idea that a sequence of essentially random or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence In Measure
Convergence in measure is either of two distinct mathematical concepts both of which generalize the concept of convergence in probability. Definitions Let f, f_n\ (n \in \mathbb N): X \to \mathbb R be measurable functions on a measure space (X, \Sigma, \mu). The sequence f_n is said to converge globally in measure to f if for every \varepsilon > 0, :\lim_ \mu(\) = 0, and to converge locally in measure to f if for every \epsilon>0 and every F \in \Sigma with \mu (F) 0 there exists ''F'' in the family such that \mu(G\setminus F)<\varepsilon. When , we may consider only one metric , so the topology of convergence in finite measure is metrizable. If is an arbitrary measure finite or not, then : still defines a metric that generates the global convergence in measure.Vladimir I. Bogachev, Measure Theory Vol. I, Springer Science & Business Media, 2007 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul-André Meyer
Paul-André Meyer (21 August 1934 – 30 January 2003) was a French mathematician, who played a major role in the development of the general theory of stochastic processes. He worked at the Institut de Recherche Mathématique (IRMA) in Strasbourg and is known as the founder of the 'Strasbourg school' in stochastic analysis. Biography Meyer was born in 1934 in Boulogne, a suburb of Paris. His family fled from France in 1940 and sailed to Argentina, settling in Buenos Aires, where Paul-André attended a French school. He returned to Paris in 1946 and entered the Lycée Janson de Sailly, where he first encountered advanced mathematics through his teacher, M Heilbronn. He entered the École Normale Supérieure in 1954 where he studied mathematics. There, he attended lectures on probability by Michel Loève, a former disciple of Paul Lévy who had come from Berkeley to spend a year in Paris. These lectures triggered Meyer's interest in the theory of stochastic processes, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Jean De La Vallée-Poussin
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was "free man". The Old English descendant of this word was '' Ċearl'' or ''Ċeorl'', as the name of King Cearl of Mercia, that disappeared after the Norman conquest of England. The name was notably borne by Charlemagne (Charles the Great), and was at the time Latinized as ''Karolus'' (as in ''Vita Karoli Magni''), later also as '' Carolus''. Some Germanic languages, for example Dutch and German, have retained the word in two separate senses. In the particular case of Dutch, ''Karel'' refers to the given name, whereas the noun ''kerel'' means "a bloke, fellow, man". Etymology The name's etymology is a Common Germanic noun ''*karilaz'' meaning "free man", which survives in English as churl (< Old English ''ċeorl''), which developed its depr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of this article will deal with this case, which is one of the concepts of functional analysis. One may call subsets of a topological vector space weakly closed (respectively, weakly compact, etc.) if they are closed (respectively, compact, etc.) with respect to the weak topology. Likewise, functions are sometimes called weakly continuous (respectively, weakly differentiable, weakly analytic, etc.) if they are continuous (respectively, differentiable, analytic, etc.) with respect to the weak topology. History Starting in the early 1900s, David Hilbert and Marcel Riesz made extensive use of weak convergence. The early pioneers o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a Hausdorff space is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is ''not'' necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. Every compact subset of a (possibly non-Hausdorff) topological vector space is complete and relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Billy James Pettis
Billy James Pettis (1913 – 14 April 1979), was an American mathematician, known for his contributions to functional analysis. See also * Dunford–Pettis property * Dunford–Pettis theorem *Milman–Pettis theorem *Orlicz–Pettis theorem *Pettis integral In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel Gelfand, Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting ... * Pettis theorem References *Graves, William H.; Davis, Robert L.; Wright, Fred B., ''Introduction''. In: Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979), pp. vii—ix, Contemporary Mathematics 2, Amer. Math. Soc., Providence, R.I., 1980. (This is an introduction to the collection of papers dedicated to the memory of B. J. Pettis.) External links *A Guide to the B. J. Pettis Papers, 1938-198 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Integrability
In mathematics, uniform integrability is an important concept in real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ..., functional analysis and measure theory, and plays a vital role in the theory of Martingale (probability theory), martingales. Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in L_1 which is central in Dominated convergence theorem, dominated convergence. Several textbooks on real analysis and measure theory use the following definition: Definition A: Let (X,\mathfrak, \mu) be a positive measure space. A set \Phi\subset L^1(\mu) is called uniformly integrable if \sup_\, f\, _0 there corresponds a \delta>0 such that : \int_E , f, \, d\mu 0 such that : \sup_\int_A, f, \, d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
